• Mar 18th 2013, 08:29 AM
tomkoolen
Hello everyone,

I'm having a bit of trouble understanding the following proof of this problem:
Attachment 27586"Prove that ABFE is a chordal quadrileteral."

Proof given by solutions book:

"Angle A = 1/2*arc CD - 1/2*arc ED = 90o- 1/2*arc ED.
Angle EFC = 1/2*arc EC = 1/2*(arc CD - arc ED) = 90o- 1/2*arc ED.
=> Angle A = Angle EFC.
Angle A + Angle EFB = Angle EFC + Angle EFB = 180o.
Thus ABFE is a chordal quadrileteral."

I do understand the logic of the proof and the whole conclusion, however I am stuck at the beginning, where the angles are given as arcs. Could anybody explain to me how this is done?

Tom
• Mar 18th 2013, 08:35 AM
Plato
Quote:

Originally Posted by tomkoolen
Hello everyone,

I'm having a bit of trouble understanding the following proof of this problem:
Attachment 27586"Prove that ABFE is a chordal quadrileteral."

Proof given by solutions book:

"Angle A = 1/2*arc CD - 1/2*arc ED = 90o- 1/2*arc ED.
Angle EFC = 1/2*arc EC = 1/2*(arc CD - arc ED) = 90o- 1/2*arc ED.
=> Angle A = Angle EFC.
Angle A + Angle EFB = Angle EFC + Angle EFB = 180o.
Thus ABFE is a chordal quadrileteral."

I do understand the logic of the proof and the whole conclusion, however I am stuck at the beginning, where the angles are given as arcs. Could anybody explain to me how this is done?

Tom

In a circle an arc angle is the length of the arc divided by the radius of the circle (formula $\displaystyle s=r\theta$). Equivalently, an arc angle is the measure of the "central" angle; i.e. the angle from the center of the circle with endpoints the endpoints of the arc. Here's the theorem which is the first line of your proof: